Bell diagonal state

The Bell diagonal state is defined as the probabilistic mixture of Bell states:

: $|\backslash phi^+\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_A\; \backslash otimes\; |0\backslash rangle\_B\; +\; |1\backslash rangle\_A\; \backslash otimes\; |1\backslash rangle\_B)$

: $|\backslash phi^-\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_A\; \backslash otimes\; |0\backslash rangle\_B\; -\; |1\backslash rangle\_A\; \backslash otimes\; |1\backslash rangle\_B)$

: $|\backslash psi^+\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_A\; \backslash otimes\; |1\backslash rangle\_B\; +\; |1\backslash rangle\_A\; \backslash otimes\; |0\backslash rangle\_B)$

: $|\backslash psi^-\backslash rangle\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\}\}\; (|0\backslash rangle\_A\; \backslash otimes\; |1\backslash rangle\_B\; -\; |1\backslash rangle\_A\; \backslash otimes\; |0\backslash rangle\_B)$

In density operator form, a Bell diagonal state is defined as

$\backslash varrho^\{Bell\}=p\_1|\backslash phi^+\backslash rangle\; \backslash langle\; \backslash phi^+|+p\_2|\backslash phi^-\backslash rangle\backslash langle\; \backslash phi^-|+p\_3|\backslash psi^+\backslash rangle\backslash langle\; \backslash psi^+|+p\_4|\backslash psi^-\backslash rangle\backslash langle\backslash psi^-|$

where $p\_1,p\_2,p\_3,p\_4$ is a probability distribution. Since $p\_1+p\_2+p\_3+p\_4=1$, a Bell diagonal state is determined by three real parameters. The maximum probability of a Bell diagonal state is defined as $p\_\{max\}=\backslash max\backslash \{p\_1,p\_2,p\_3,p\_4\backslash \}$.

1. A Bell-diagonal state is separable if all the probabilities are less or equal to 1/2, i.e., $p\_\backslash text\{max\}\backslash leq\; 1/2$.{{Cite journal |last1=Horodecki |first1=Ryszard |last2=Horodecki |first2=Michal/ |date=1996-09-01 |title=Information-theoretic aspects of inseparability of mixed states |url=https://link.aps.org/doi/10.1103/PhysRevA.54.1838 |journal=Physical Review A |volume=54 |issue=3 |pages=1838–1843 |doi=10.1103/PhysRevA.54.1838|pmid=9913669 |arxiv=quant-ph/9607007 |bibcode=1996PhRvA..54.1838H |s2cid=2340228 }}

2. Many entanglement measures have a simple formulas for entangled Bell-diagonal states:

Relative entropy of entanglement: $S\_r=1-h(p\_\backslash text\{max\})$,{{Cite journal |last1=Vedral |first1=V. |last2=Plenio |first2=M. B. |last3=Rippin |first3=M. A. |last4=Knight |first4=P. L. |date=1997-03-24 |title=Quantifying Entanglement |url=https://link.aps.org/doi/10.1103/PhysRevLett.78.2275 |journal=Physical Review Letters |volume=78 |issue=12 |pages=2275–2279 |doi=10.1103/PhysRevLett.78.2275|arxiv=quant-ph/9702027 |bibcode=1997PhRvL..78.2275V |hdl=10044/1/300 |s2cid=16118336 }} where $h$ is the binary entropy function.

Entanglement of formation: $E\_f=h(\backslash frac\{1\}\{2\}+\backslash sqrt\{p\_\backslash text\{max\}(1-p\_\backslash text\{max\})\})$,where $h$ is the binary entropy function.

Negativity: $N=p\_\backslash text\{max\}-1/2$

Log-negativity: $E\_N=\backslash log(2\; p\_\backslash text\{max\}\; )$

3. Any 2-qubit state where the reduced density matrices are maximally mixed, $\backslash rho\_A=\backslash rho\_B=I/2$, is Bell-diagonal in some local basis. Viz., there exist local unitaries $U=U\_1\backslash otimes\; U\_2$ such that $U\backslash rho\; U^\{\backslash dagger\}$ is Bell-diagonal.

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Category:Quantum information science

Category:Quantum states